Usaldusvahemik

.title[
# Usaldusvahemik
]
.author[
### Indrek Soidla
]
.institute[
### Tartu Ülikool
]
.date[
### 2023/09/23 (updated: 2023-10-04)
]

---

### Laeme andmed sisse, eraldame Eesti andmed

```r
library(haven)
library(tidyverse)
r8 <- read_spss("data/ESS8e02_2.sav")

ee8 <- r8 |>
  filter(cntry == "EE") |>
  select(lkredcc, region, pspwght)
```

---

### Kontrollime tunnuse `lkredcc` jaotusparameetreid

```r
library(summarytools)
descr(ee8$lkredcc)
```

```
## Descriptive Statistics  
## ee8$lkredcc  
## Label: Imagine large numbers of people limit energy use, how likely reduce climate change  
## N: 2019  
## 
##                     lkredcc
## ----------------- ---------
##              Mean      4.70
##           Std.Dev      2.43
##               Min      0.00
##                Q1      3.00
##            Median      5.00
##                Q3      6.00
##               Max     10.00
##               MAD      2.97
##               IQR      3.00
##                CV      0.52
##          Skewness     -0.13
##       SE.Skewness      0.06
##          Kurtosis     -0.54
##           N.Valid   1905.00
##         Pct.Valid     94.35
```

---

### Kodeerime tunnuse `region` ümber eestikeelseks

```r
ee8$region <- dplyr::recode(as.factor(ee8$region), 
                     "EE001" = "Põhja-Eesti", 
                     "EE004" = "Lääne-Eesti",
                     "EE006" = "Kesk-Eesti",
                     "EE007" = "Kirde-Eesti",
                     "EE008" = "Lõuna-Eesti")
table(ee8$region)
```

```
## 
## Põhja-Eesti Lääne-Eesti  Kesk-Eesti Kirde-Eesti Lõuna-Eesti 
##         824         236         213         209         537
```

---

### Arvutame paketi `meantables` abiga hinnangute keskmised ja usalduspiirid regiooniti

```r
# install.packages("meantables")
library(meantables)

ee8 |>
  group_by(region) |>
  mean_table(lkredcc)
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> response_var </th>
   <th style="text-align:left;"> group_var </th>
   <th style="text-align:left;"> group_cat </th>
   <th style="text-align:right;"> n </th>
   <th style="text-align:right;"> mean </th>
   <th style="text-align:right;"> sd </th>
   <th style="text-align:right;"> sem </th>
   <th style="text-align:right;"> lcl </th>
   <th style="text-align:right;"> ucl </th>
   <th style="text-align:right;"> min </th>
   <th style="text-align:right;"> max </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> lkredcc </td>
   <td style="text-align:left;"> region </td>
   <td style="text-align:left;"> Põhja-Eesti </td>
   <td style="text-align:right;"> 792 </td>
   <td style="text-align:right;"> 4.98 </td>
   <td style="text-align:right;"> 2.34 </td>
   <td style="text-align:right;"> 0.0830960 </td>
   <td style="text-align:right;"> 4.82 </td>
   <td style="text-align:right;"> 5.15 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> lkredcc </td>
   <td style="text-align:left;"> region </td>
   <td style="text-align:left;"> Lääne-Eesti </td>
   <td style="text-align:right;"> 232 </td>
   <td style="text-align:right;"> 4.57 </td>
   <td style="text-align:right;"> 2.41 </td>
   <td style="text-align:right;"> 0.1582965 </td>
   <td style="text-align:right;"> 4.26 </td>
   <td style="text-align:right;"> 4.88 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> lkredcc </td>
   <td style="text-align:left;"> region </td>
   <td style="text-align:left;"> Kesk-Eesti </td>
   <td style="text-align:right;"> 181 </td>
   <td style="text-align:right;"> 4.53 </td>
   <td style="text-align:right;"> 2.67 </td>
   <td style="text-align:right;"> 0.1986049 </td>
   <td style="text-align:right;"> 4.14 </td>
   <td style="text-align:right;"> 4.92 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> lkredcc </td>
   <td style="text-align:left;"> region </td>
   <td style="text-align:left;"> Kirde-Eesti </td>
   <td style="text-align:right;"> 185 </td>
   <td style="text-align:right;"> 4.08 </td>
   <td style="text-align:right;"> 2.43 </td>
   <td style="text-align:right;"> 0.1784336 </td>
   <td style="text-align:right;"> 3.73 </td>
   <td style="text-align:right;"> 4.43 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> lkredcc </td>
   <td style="text-align:left;"> region </td>
   <td style="text-align:left;"> Lõuna-Eesti </td>
   <td style="text-align:right;"> 515 </td>
   <td style="text-align:right;"> 4.61 </td>
   <td style="text-align:right;"> 2.46 </td>
   <td style="text-align:right;"> 0.1082654 </td>
   <td style="text-align:right;"> 4.40 </td>
   <td style="text-align:right;"> 4.82 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
</tbody>
</table>

---

### Punktdiagramm

```r
lk_95 <- ee8 |>
  group_by(region) |>
  mean_table(lkredcc)
```

```r
ggplot(lk_95, 
       aes(x = group_cat, 
                  y = mean)) +
  geom_point()
```
]

.pull-right[
![](data:image/png;base64,#km_04_usaldusvahemik_kood_files/figure-html/unnamed-chunk-6-1.png)
]

---

### Vahemikdiagramm

```r
ggplot(lk_95, 
       aes(x = group_cat, 
           y = mean)) +
  geom_point() +
  geom_errorbar(
    aes(ymin = lcl, 
        ymax = ucl), 
    width = 0.1)
```
]

.pull-right[
![](data:image/png;base64,#km_04_usaldusvahemik_kood_files/figure-html/unnamed-chunk-7-1.png)
]

---

### Millest sõltub usaldusvahemiku laius?

```r
ee8 |>
  group_by(region) |>
  mean_table(lkredcc) |>
  mutate(laius = ucl - lcl) |>
  select(group_cat, n, mean, sd, laius)
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> group_cat </th>
   <th style="text-align:right;"> n </th>
   <th style="text-align:right;"> mean </th>
   <th style="text-align:right;"> sd </th>
   <th style="text-align:right;"> laius </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Põhja-Eesti </td>
   <td style="text-align:right;"> 792 </td>
   <td style="text-align:right;"> 4.98 </td>
   <td style="text-align:right;"> 2.34 </td>
   <td style="text-align:right;"> 0.33 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Lääne-Eesti </td>
   <td style="text-align:right;"> 232 </td>
   <td style="text-align:right;"> 4.57 </td>
   <td style="text-align:right;"> 2.41 </td>
   <td style="text-align:right;"> 0.62 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Kesk-Eesti </td>
   <td style="text-align:right;"> 181 </td>
   <td style="text-align:right;"> 4.53 </td>
   <td style="text-align:right;"> 2.67 </td>
   <td style="text-align:right;"> 0.78 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Kirde-Eesti </td>
   <td style="text-align:right;"> 185 </td>
   <td style="text-align:right;"> 4.08 </td>
   <td style="text-align:right;"> 2.43 </td>
   <td style="text-align:right;"> 0.70 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Lõuna-Eesti </td>
   <td style="text-align:right;"> 515 </td>
   <td style="text-align:right;"> 4.61 </td>
   <td style="text-align:right;"> 2.46 </td>
   <td style="text-align:right;"> 0.42 </td>
  </tr>
</tbody>
</table>

---

### Kuidas sõltub usaldusvahemiku laius usaldusnivoost?

Arvutame usalduspiirid ka usaldusnivoodel 99% ja 90%

```r
lk_99 <- ee8 |>
  group_by(region) |>
  mean_table(lkredcc, t_prob = 0.995)

lk_90 <- ee8 |>
  group_by(region) |>
  mean_table(lkredcc, t_prob = 0.95)
```

---

### Usaldusvahemikud eri usaldusnivoodel

```r
ggplot(lk_95, aes(x = group_cat, y = mean)) +
  geom_point() +
* geom_errorbar(data = lk_99, aes(ymin = lcl, ymax = ucl),
*               width = 0.05, colour = "blue") +
  geom_errorbar(aes(ymin = lcl, ymax = ucl), width = 0.1) +
* geom_errorbar(data = lk_90, aes(ymin = lcl, ymax = ucl),
*               width = 0.15, colour = "red")
```

---

### Usaldusvahemikud eri usaldusnivoodel

![](data:image/png;base64,#km_04_usaldusvahemik_kood_files/figure-html/unnamed-chunk-11-1.png)

---

### Uurime lähemalt usaldusvahemikke usaldusnivool 95%

```r
ggplot(lk_95, 
       aes(x = group_cat, 
           y = mean)) +
  geom_point() +
  geom_errorbar(
    aes(ymin = lcl, 
        ymax = ucl), 
    width = 0.1)
```
]

.pull-right[
![](data:image/png;base64,#km_04_usaldusvahemik_kood_files/figure-html/unnamed-chunk-13-1.png)
]

---

### Harjutus 3

```r
ee8a <- r8 %>% 
  filter(cntry == "EE") %>% 
  select(gvsrdcc, edulvlb, pspwght)

ee8a <- ee8a %>% 
  mutate(har = case_when(edulvlb <= 213 ~ "Kuni põhiharidus",
                         edulvlb == 313 ~ "Keskharidus",
                         edulvlb > 213 & edulvlb < 600 ~ "Kutseharidus",
                         edulvlb >= 600 ~ "Kõrgharidus"))

library(forcats)
ee8a$har <- fct_relevel(ee8a$har, 
                        "Kuni põhiharidus", 
                        "Kutseharidus", 
                        "Keskharidus", 
                        "Kõrgharidus")
```

---

### Harjutus 3

```r
gs_uv <- ee8a |>
  drop_na(har) |>
  group_by(har) |>
  mean_table(gvsrdcc)
```

```r
ggplot(gs_uv, 
       aes(x = group_cat, 
                  y = mean)) +
  geom_point() +
  geom_errorbar(
    aes(ymin = lcl, 
        ymax = ucl), 
    width = 0.1
    )
```
]

.pull-right[
![](data:image/png;base64,#km_04_usaldusvahemik_kood_files/figure-html/unnamed-chunk-16-1.png)
]

---

# Keskmiste vahe usalduspiirid

---

### Arvutame keskmiste vahede usalduspiirid

```r
ee8 %>%   filter(region == "Kirde-Eesti" | region == "Kesk-Eesti") %>% 
  t.test(lkredcc ~ region, data = .)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  lkredcc by region
## t = 1.6829, df = 359.03, p-value = 0.09327
## alternative hypothesis: true difference in means between group Kesk-Eesti and group Kirde-Eesti is not equal to 0
## 95 percent confidence interval:
##  -0.07575068  0.97436200
## sample estimates:
##  mean in group Kesk-Eesti mean in group Kirde-Eesti 
##                  4.530387                  4.081081
```

---

### Arvutame keskmiste vahede usalduspiirid

```r
ee8 %>% 
  filter(region == "Kirde-Eesti" | region == "Lääne-Eesti") %>% 
  t.test(lkredcc ~ region, data = .)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  lkredcc by region
## t = 2.0454, df = 393.47, p-value = 0.04148
## alternative hypothesis: true difference in means between group Lääne-Eesti and group Kirde-Eesti is not equal to 0
## 95 percent confidence interval:
##  0.01893274 0.95683613
## sample estimates:
## mean in group Lääne-Eesti mean in group Kirde-Eesti 
##                  4.568966                  4.081081
```

---

### Harjutus 4

```r
ee8a %>% 
  filter(har == "Kuni põhiharidus" | har == "Kutseharidus") %>% 
  t.test(gvsrdcc ~ har, data = .)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  gvsrdcc by har
## t = -2.0146, df = 598.41, p-value = 0.04439
## alternative hypothesis: true difference in means between group Kuni põhiharidus and group Kutseharidus is not equal to 0
## 95 percent confidence interval:
##  -0.612921963 -0.007803079
## sample estimates:
## mean in group Kuni põhiharidus     mean in group Kutseharidus 
##                       4.312704                       4.623066
```

---

### Harjutus 4

```r
ee8a %>% 
  filter(har == "Kuni põhiharidus" | har == "Keskharidus") %>% 
  t.test(gvsrdcc ~ har, data = .)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  gvsrdcc by har
## t = -2.4051, df = 654.21, p-value = 0.01644
## alternative hypothesis: true difference in means between group Kuni põhiharidus and group Keskharidus is not equal to 0
## 95 percent confidence interval:
##  -0.76908464 -0.07773041
## sample estimates:
## mean in group Kuni põhiharidus      mean in group Keskharidus 
##                       4.312704                       4.736111
```

---

### Harjutus 4

```r
ee8a %>% 
  filter(har == "Kutseharidus" | har == "Keskharidus") %>% 
  t.test(gvsrdcc ~ har, data = .)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  gvsrdcc by har
## t = -0.75807, df = 722.09, p-value = 0.4487
## alternative hypothesis: true difference in means between group Kutseharidus and group Keskharidus is not equal to 0
## 95 percent confidence interval:
##  -0.4058079  0.1797179
## sample estimates:
## mean in group Kutseharidus  mean in group Keskharidus 
##                   4.623066                   4.736111
```

---

### Harjutus 4

```r
ee8a %>% 
  filter(har == "Keskharidus" | har == "Kõrgharidus") %>% 
  t.test(gvsrdcc ~ har, data = .)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  gvsrdcc by har
## t = -2.4046, df = 712.09, p-value = 0.01645
## alternative hypothesis: true difference in means between group Keskharidus and group Kõrgharidus is not equal to 0
## 95 percent confidence interval:
##  -0.65796689 -0.06647137
## sample estimates:
## mean in group Keskharidus mean in group Kõrgharidus 
##                  4.736111                  5.098330
```

---

# Usaldusvahemike arvutamine kaaludega

---

### Usaldusvahemike arvutamine kaaludega

- Eelnevad näited olid lihtsuse mõttes tehtud ilma kaaludeta, täpsemate tulemuste saamiseks oleks tarvis ESS-i andmete puhul kasutada ka järelkihistamiskaale

- Selleks on tarvis paketi survey abi

```r
# install.packages("survey")
library(survey)

ee8w <- svydesign(id = ~1, data = ee8, weights = ~pspwght)
```

---

### Arvutame tunnuse lkredcc kaalutud keskmise ja usalduspiirid

```r
svymean(~lkredcc, design = ee8w, na.rm = TRUE)
```

```
##           mean     SE
## lkredcc 4.7346 0.0559
```

```r
svymean(~lkredcc, design = ee8w, na.rm = TRUE) %>% 
  confint()
```

```
##            2.5 %   97.5 %
## lkredcc 4.625124 4.844147
```

---

### Arvutame tunnuse lkredcc kaalutud keskmised ja usalduspiirid regioonides

```r
svyby(~lkredcc, ~region, design = ee8w, FUN = svymean, 
      na.rm = TRUE, vartype = c("se", "ci"))
```

```
##                  region  lkredcc        se     ci_l     ci_u
## Põhja-Eesti Põhja-Eesti 5.006927 0.0831522 4.843952 5.169903
## Lääne-Eesti Lääne-Eesti 4.603206 0.1554963 4.298439 4.907973
## Kesk-Eesti   Kesk-Eesti 4.509991 0.1950830 4.127636 4.892347
## Kirde-Eesti Kirde-Eesti 4.150544 0.1833316 3.791220 4.509867
## Lõuna-Eesti Lõuna-Eesti 4.631699 0.1097541 4.416585 4.846813
```

---

### Teeme sõltumatute kogumite t-testi kaalutud andmetega

Loome iga regiooni jaoks eraldi andmestikud

```r
# install.packages("weights")
library(weights)

lk_ki <- ee8 %>% 
  filter(region == "Kirde-Eesti")

lk_ke <- ee8 %>% 
  filter(region == "Kesk-Eesti")

lk_le <- ee8 %>% 
  filter(region == "Lõuna-Eesti")
```

---

### Teeme sõltumatute kogumite t-testi kaalutud andmetega

```r
wtd.t.test(lk_ki$lkredcc, lk_ke$lkredcc, 
           weight = lk_ki$pspwght, weighty = lk_ke$pspwght)
```

```
## Warning in wtd.t.test(lk_ki$lkredcc, lk_ke$lkredcc, weight = lk_ki$pspwght, :
## Treating data for x and y separately because they are of different lengths
```

```
## $test
## [1] "Two Sample Weighted T-Test (Welch)"
## 
## $coefficients
##     t.value          df     p.value 
##  -1.3606133 363.2974703   0.1744794 
## 
## $additional
## Difference     Mean.x     Mean.y   Std. Err 
## -0.3594476  4.1505435  4.5099911  0.2641806
```

---

### Teeme sõltumatute kogumite t-testi kaalutud andmetega

```r
wtd.t.test(lk_ki$lkredcc, lk_le$lkredcc, 
           weight = lk_ki$pspwght, weighty = lk_le$pspwght)
```

```
## Warning in wtd.t.test(lk_ki$lkredcc, lk_le$lkredcc, weight = lk_ki$pspwght, :
## Treating data for x and y separately because they are of different lengths
```

```
## $test
## [1] "Two Sample Weighted T-Test (Welch)"
## 
## $coefficients
##      t.value           df      p.value 
##  -2.30706395 329.45292273   0.02167097 
## 
## $additional
## Difference     Mean.x     Mean.y   Std. Err 
## -0.4811557  4.1505435  4.6316992  0.2085576
```

---

### `wtd.t.test` ei anna kaalutud keskmiste vahe usalduspiire

- Aga need saab ise arvutada standardvea põhjal

- Omistame testitulemused uuele objektile `t_test_ki_le`

```r
t_test_ki_le <- wtd.t.test(lk_ki$lkredcc, lk_le$lkredcc, 
                           weight = lk_ki$pspwght, 
                           weighty = lk_le$pspwght)
```

---

### Arvutame keskmiste vahe usalduspiirid

```r
str(t_test_ki_le)
```

```
## List of 3
##  $ test        : chr "Two Sample Weighted T-Test (Welch)"
##  $ coefficients: Named num [1:3] -2.3071 329.4529 0.0217
##   ..- attr(*, "names")= chr [1:3] "t.value" "df" "p.value"
##  $ additional  : Named num [1:4] -0.481 4.151 4.632 0.209
##   ..- attr(*, "names")= chr [1:4] "Difference" "Mean.x" "Mean.y" "Std. Err"
```

- Alumine usalduspiir usaldusnivool 95%

```r
t_test_ki_le$additional[1] - 1.96 * t_test_ki_le$additional[4]
```

```
## Difference 
## -0.8899285
```

- Ülemine usalduspiir usaldusnivool 95%

```r
t_test_ki_le$additional[1] + 1.96 * t_test_ki_le$additional[4]
```

```
##  Difference 
## -0.07238281
```

---

### Harjutus 5

```r
ee8w <- svydesign(id = ~1, data = ee8a, weights = ~pspwght)

svyby(~gvsrdcc, ~har, design = ee8w, FUN = svymean, 
      na.rm = TRUE, vartype = c("se", "ci"))
```

```
##                               har  gvsrdcc         se     ci_l     ci_u
## Kuni põhiharidus Kuni põhiharidus 4.362288 0.12484189 4.117603 4.606974
## Kutseharidus         Kutseharidus 4.610041 0.08721434 4.439104 4.780978
## Keskharidus           Keskharidus 4.755176 0.12423536 4.511679 4.998673
## Kõrgharidus           Kõrgharidus 5.094212 0.08988150 4.918048 5.270377
```

```r
gs_uv_w <- svyby(~gvsrdcc, ~har, design = ee8w, FUN = svymean, 
      na.rm = TRUE, vartype = c("se", "ci"))
```

```r
ggplot(gs_uv_w, aes(x = har, y = gvsrdcc)) +
  geom_point(stat = "identity") +
  geom_errorbar(aes(ymin = ci_l, ymax = ci_u), width = 0.1) +
  labs(title = "Vahemikdiagramm kaalutud andmetega") +
  ylim(4, 5.5)
```

---

### Harjutus 5

.pull-left[
![](data:image/png;base64,#km_04_usaldusvahemik_kood_files/figure-html/unnamed-chunk-35-1.png)

]

.pull-right[
![](data:image/png;base64,#km_04_usaldusvahemik_kood_files/figure-html/unnamed-chunk-36-1.png)
]

---

# Osakaalu usaldusvahemik

---

### Osakaalu (protsentnäitaja) usalduspiirid

`$$p >= \hat{p} - z_{1 - \frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$`

`$$p <= \hat{p} + z_{1 - \frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$`

- `$p$` -- osakaal populatsioonis

- `$\hat{p}$` -- osakaal valimis

- `$z_{1 - \frac{\alpha}{2}}$` -- standardiseeritud normaaljaotuse `$1 - \frac{\alpha}{2}$`-kvantiil, kus `$\alpha$` on vea tõenäosuse piir

- nt usaldusnivool 90% `$\alpha = 10\%$`, vastav `$z_{1 - \frac{\alpha}{2}} = 1.64$`
  - nt usaldusnivool 95% `$\alpha = 5\%$`, vastav `$z_{1 - \frac{\alpha}{2}} = 1.96$`
  - nt usaldusnivool 99% `$\alpha = 1\%$`, vastav `$z_{1 - \frac{\alpha}{2}} = 2.58$`

- `$n$` -- valimi suurus

- Mida lähemal osakaal 50%-le, seda laiem on usaldusvahemik

---

### Osakaalu usaldusvahemiku laius

- Kuidas sõltub osakaalu usaldusvahemik osakaalu (protsentnäitaja) suurusest?

- Näide valimi n = 1000 kohta

```r
p <- seq(0, 1, 0.01)

dat <- data.frame(p)

dat$veapiir <- 1.96 * sqrt(p * (1 - p) / 1000)

knitr::kable(dat, format = 'html')
```

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> p </th>
   <th style="text-align:right;"> veapiir </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> 0.0000000 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;"> 0.0061670 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;"> 0.0086773 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.03 </td>
   <td style="text-align:right;"> 0.0105731 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.04 </td>
   <td style="text-align:right;"> 0.0121457 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> 0.0135084 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.06 </td>
   <td style="text-align:right;"> 0.0147196 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.07 </td>
   <td style="text-align:right;"> 0.0158142 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.08 </td>
   <td style="text-align:right;"> 0.0168149 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.09 </td>
   <td style="text-align:right;"> 0.0177377 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.10 </td>
   <td style="text-align:right;"> 0.0185942 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.11 </td>
   <td style="text-align:right;"> 0.0193931 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.12 </td>
   <td style="text-align:right;"> 0.0201413 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.13 </td>
   <td style="text-align:right;"> 0.0208443 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.14 </td>
   <td style="text-align:right;"> 0.0215065 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.15 </td>
   <td style="text-align:right;"> 0.0221315 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.16 </td>
   <td style="text-align:right;"> 0.0227225 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.17 </td>
   <td style="text-align:right;"> 0.0232820 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.18 </td>
   <td style="text-align:right;"> 0.0238122 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.19 </td>
   <td style="text-align:right;"> 0.0243151 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.20 </td>
   <td style="text-align:right;"> 0.0247923 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.21 </td>
   <td style="text-align:right;"> 0.0252452 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.22 </td>
   <td style="text-align:right;"> 0.0256753 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.23 </td>
   <td style="text-align:right;"> 0.0260835 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.24 </td>
   <td style="text-align:right;"> 0.0264709 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.25 </td>
   <td style="text-align:right;"> 0.0268384 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.26 </td>
   <td style="text-align:right;"> 0.0271868 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.27 </td>
   <td style="text-align:right;"> 0.0275169 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.28 </td>
   <td style="text-align:right;"> 0.0278292 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.29 </td>
   <td style="text-align:right;"> 0.0281245 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.30 </td>
   <td style="text-align:right;"> 0.0284031 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.31 </td>
   <td style="text-align:right;"> 0.0286656 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.32 </td>
   <td style="text-align:right;"> 0.0289125 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.33 </td>
   <td style="text-align:right;"> 0.0291441 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.34 </td>
   <td style="text-align:right;"> 0.0293608 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.35 </td>
   <td style="text-align:right;"> 0.0295629 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.36 </td>
   <td style="text-align:right;"> 0.0297507 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.37 </td>
   <td style="text-align:right;"> 0.0299245 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.38 </td>
   <td style="text-align:right;"> 0.0300846 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.39 </td>
   <td style="text-align:right;"> 0.0302311 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.40 </td>
   <td style="text-align:right;"> 0.0303642 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.41 </td>
   <td style="text-align:right;"> 0.0304841 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.42 </td>
   <td style="text-align:right;"> 0.0305911 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.43 </td>
   <td style="text-align:right;"> 0.0306851 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.44 </td>
   <td style="text-align:right;"> 0.0307664 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.45 </td>
   <td style="text-align:right;"> 0.0308350 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.46 </td>
   <td style="text-align:right;"> 0.0308910 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.47 </td>
   <td style="text-align:right;"> 0.0309345 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.48 </td>
   <td style="text-align:right;"> 0.0309655 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.49 </td>
   <td style="text-align:right;"> 0.0309841 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.50 </td>
   <td style="text-align:right;"> 0.0309903 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.51 </td>
   <td style="text-align:right;"> 0.0309841 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.52 </td>
   <td style="text-align:right;"> 0.0309655 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.53 </td>
   <td style="text-align:right;"> 0.0309345 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.54 </td>
   <td style="text-align:right;"> 0.0308910 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.55 </td>
   <td style="text-align:right;"> 0.0308350 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.56 </td>
   <td style="text-align:right;"> 0.0307664 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.57 </td>
   <td style="text-align:right;"> 0.0306851 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.58 </td>
   <td style="text-align:right;"> 0.0305911 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.59 </td>
   <td style="text-align:right;"> 0.0304841 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.60 </td>
   <td style="text-align:right;"> 0.0303642 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.61 </td>
   <td style="text-align:right;"> 0.0302311 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.62 </td>
   <td style="text-align:right;"> 0.0300846 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.63 </td>
   <td style="text-align:right;"> 0.0299245 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.64 </td>
   <td style="text-align:right;"> 0.0297507 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.65 </td>
   <td style="text-align:right;"> 0.0295629 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.66 </td>
   <td style="text-align:right;"> 0.0293608 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.67 </td>
   <td style="text-align:right;"> 0.0291441 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.68 </td>
   <td style="text-align:right;"> 0.0289125 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.69 </td>
   <td style="text-align:right;"> 0.0286656 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.70 </td>
   <td style="text-align:right;"> 0.0284031 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.71 </td>
   <td style="text-align:right;"> 0.0281245 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.72 </td>
   <td style="text-align:right;"> 0.0278292 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.73 </td>
   <td style="text-align:right;"> 0.0275169 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.74 </td>
   <td style="text-align:right;"> 0.0271868 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.75 </td>
   <td style="text-align:right;"> 0.0268384 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.76 </td>
   <td style="text-align:right;"> 0.0264709 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.77 </td>
   <td style="text-align:right;"> 0.0260835 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.78 </td>
   <td style="text-align:right;"> 0.0256753 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.79 </td>
   <td style="text-align:right;"> 0.0252452 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.80 </td>
   <td style="text-align:right;"> 0.0247923 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.81 </td>
   <td style="text-align:right;"> 0.0243151 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.82 </td>
   <td style="text-align:right;"> 0.0238122 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.83 </td>
   <td style="text-align:right;"> 0.0232820 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.84 </td>
   <td style="text-align:right;"> 0.0227225 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.85 </td>
   <td style="text-align:right;"> 0.0221315 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.86 </td>
   <td style="text-align:right;"> 0.0215065 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.87 </td>
   <td style="text-align:right;"> 0.0208443 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.88 </td>
   <td style="text-align:right;"> 0.0201413 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.89 </td>
   <td style="text-align:right;"> 0.0193931 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.90 </td>
   <td style="text-align:right;"> 0.0185942 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.91 </td>
   <td style="text-align:right;"> 0.0177377 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.92 </td>
   <td style="text-align:right;"> 0.0168149 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.93 </td>
   <td style="text-align:right;"> 0.0158142 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.94 </td>
   <td style="text-align:right;"> 0.0147196 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> 0.0135084 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.96 </td>
   <td style="text-align:right;"> 0.0121457 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.97 </td>
   <td style="text-align:right;"> 0.0105731 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.98 </td>
   <td style="text-align:right;"> 0.0086773 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.99 </td>
   <td style="text-align:right;"> 0.0061670 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1.00 </td>
   <td style="text-align:right;"> 0.0000000 </td>
  </tr>
</tbody>
</table>

---

### Osakaalu usaldusvahemiku laius

```r
joonis <- ggplot(dat, aes(p, veapiir)) +
  geom_path()
```

---

### Osakaalu usaldusvahemiku laius

```r
plotly::ggplotly(joonis)
```

<div class="plotly html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-ca1b816f9de8e9f4ecc5" style="width:504px;height:504px;"></div>
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---

### Veapiir

- Pool usaldusvahemikust ehk `$z_{1 - \frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$` = veapiir

- Korrektne kasutada ainult tõenäosusliku valimiga uuringu puhul

- See eeldus kehtib igasuguse järeldava analüüsi kohta

---

### Osakaalude usaldusvahemike arvutamine

- Kasutame kahe erakondade reitingu uuringu andmeid (aastast 2020):

- Turu-uuringute AS, 12.-24. aug, n = 1003 (https://www.err.ee/1127504/turu-uuringud-reformierakonna-toetus-jai-augustis-keskerakonnale-alla)

- Turu-uuringute AS, 7.-17. sept, n = 1010, https://www.err.ee/1138657/reitingud-reformierakond-tousis-taas-populaarseimaks-parteiks).

- Näide on selles mõttes meie jaoks kehv, et reitingud on ümardatud täisarvuni, aga käsitleme siis neid harjutuse korras kui uuringuga saadud tulemusi

---

### Osakaalude usaldusvahemike arvutamine

```r
erakond <- rep(c("RE", "KE", "EKRE", "SDE", "Eesti 200", "Isamaa", 
                 "Rohelised", "Tulevik", "Muu"), 2)
reiting <- data.frame(erakond)
reiting$poolehoid <- c(0.23, 0.26, 0.20, 0.10, 0.09, 0.05, 0.02, 
                       0.01, 0.04, 0.27, 0.24, 0.16, 0.11, 0.09, 
                       0.05, 0.03, 0.03, 0.02)
reiting$kuu <- c(rep("August", 9), rep("September", 9))
reiting
```

```
##      erakond poolehoid       kuu
## 1         RE      0.23    August
## 2         KE      0.26    August
## 3       EKRE      0.20    August
## 4        SDE      0.10    August
## 5  Eesti 200      0.09    August
## 6     Isamaa      0.05    August
## 7  Rohelised      0.02    August
## 8    Tulevik      0.01    August
## 9        Muu      0.04    August
## 10        RE      0.27 September
## 11        KE      0.24 September
## 12      EKRE      0.16 September
## 13       SDE      0.11 September
## 14 Eesti 200      0.09 September
## 15    Isamaa      0.05 September
## 16 Rohelised      0.03 September
## 17   Tulevik      0.03 September
## 18       Muu      0.02 September
```

---

### Vaatame tulemusi ka joonisel

```r
ggplot(reiting, aes(x = erakond, y = poolehoid, fill = kuu)) +
  geom_bar(stat = "identity", position = "dodge", width = 0.6) +
  scale_x_discrete(limits = reiting$erakond[1:9]) +
  labs(x = "Erakond",
       y = "Osakaal") +
  theme_light() 
```

---

### Vaatame tulemusi ka joonisel

---

### Osakaalude usaldusvahemike arvutamine

- Uudise pealkiri oli, et RE tõusis taas populaarseimaks parteiks

- Kas saame siiski uuringu andmetel väita, et muutus RE ja KE poolehoiuprotsentides esineb ka populatsioonis (eeldades, et täisarvulised protsendid on täpsed)?

- Arvutame usalduspiirid ja lisame joonisele

---

### Osakaalude usaldusvahemike arvutamine

- Kasutame usaldusvahemike arvutamiseks funktsiooni `MultinomCI`, mis eeldab indiviidide arve, mitte protsentjaotust

- Seega arvutame kõigepealt reitingute põhjal vastajate arvud (st kui palju vastajaid mingit erakonda eelistavad)

```r
reiting <- reiting %>% 
  mutate(poolehoid_n = ifelse(kuu == "August", 
                              poolehoid * 1003, 
                              poolehoid * 1010) %>% 
           round())
```

---

### Osakaalude usaldusvahemike arvutamine

- Arvutame usalduspiirid, kasutades funktsiooni `MultinomCI` paketist `DescTools`

- Reitingute usalduspiirid on vaja arvutada augusti ja septembri kohta eraldi, sest eri kuude reitingud moodustavad omaette tervikud (ühe kuu reitingud moodustavad kokku 100%)

```r
# install.packages("DescTools")
library(DescTools)

poolehoid_aug <- reiting %>% 
  filter(kuu == "August") %>% 
  pull(poolehoid_n) %>% 
  MultinomCI(conf.level = 0.95, method = "wilson")

poolehoid_sept <- reiting %>% 
  filter(kuu == "September") %>% 
  pull(poolehoid_n) %>% 
  MultinomCI(conf.level = 0.95, method = "wilson")
```

---

### Paneme reitingute usalduspiirid kokku üheks tabeliks objekti `poolehoid_ci`

```r
poolehoid_ci <- rbind(poolehoid_aug, poolehoid_sept) %>% 
  as.data.frame()
poolehoid_ci
```

```
##           est      lwr.ci     upr.ci
## 1  0.23030907 0.205311231 0.25736485
## 2  0.26021934 0.234017535 0.28825085
## 3  0.20039880 0.176789596 0.22629418
## 4  0.09970090 0.082659353 0.11979701
## 5  0.08973081 0.073573675 0.10901859
## 6  0.04985045 0.038015603 0.06512026
## 7  0.01994018 0.012944740 0.03059882
## 8  0.00997009 0.005424455 0.01825500
## 9  0.03988036 0.029422300 0.05384946
## 10 0.27056492 0.244061881 0.29880833
## 11 0.23984143 0.214513526 0.26714276
## 12 0.16055500 0.139196600 0.18448827
## 13 0.11000991 0.092162180 0.13081592
## 14 0.09018831 0.074033075 0.10945217
## 15 0.04955401 0.037788342 0.06473655
## 16 0.02973241 0.020904886 0.04212715
## 17 0.02973241 0.020904886 0.04212715
## 18 0.01982161 0.012867550 0.03041806
```

---

### Lisame algsest andmestikust ka erakondade nimed ja kuud

```r
poolehoid_ci$erakond <- reiting$erakond
poolehoid_ci$kuu <- reiting$kuu
poolehoid_ci
```

```
##           est      lwr.ci     upr.ci   erakond       kuu
## 1  0.23030907 0.205311231 0.25736485        RE    August
## 2  0.26021934 0.234017535 0.28825085        KE    August
## 3  0.20039880 0.176789596 0.22629418      EKRE    August
## 4  0.09970090 0.082659353 0.11979701       SDE    August
## 5  0.08973081 0.073573675 0.10901859 Eesti 200    August
## 6  0.04985045 0.038015603 0.06512026    Isamaa    August
## 7  0.01994018 0.012944740 0.03059882 Rohelised    August
## 8  0.00997009 0.005424455 0.01825500   Tulevik    August
## 9  0.03988036 0.029422300 0.05384946       Muu    August
## 10 0.27056492 0.244061881 0.29880833        RE September
## 11 0.23984143 0.214513526 0.26714276        KE September
## 12 0.16055500 0.139196600 0.18448827      EKRE September
## 13 0.11000991 0.092162180 0.13081592       SDE September
## 14 0.09018831 0.074033075 0.10945217 Eesti 200 September
## 15 0.04955401 0.037788342 0.06473655    Isamaa September
## 16 0.02973241 0.020904886 0.04212715 Rohelised September
## 17 0.02973241 0.020904886 0.04212715   Tulevik September
## 18 0.01982161 0.012867550 0.03041806       Muu September
```

---

### Paneme reitingud koos usalduspiiridega joonisele

```r
library(scales)
ggplot(poolehoid_ci, aes(x = erakond, y = est, fill = kuu)) +
  geom_bar(stat = "identity", position = "dodge", width = 0.9) +
  geom_errorbar(aes(ymin = lwr.ci, ymax = upr.ci), position = position_dodge2(width = 0.6, padding = 0.6), color = "black") +
  scale_y_continuous(labels = scales::percent_format(accuracy = 1L)) +
  scale_x_discrete(limits = poolehoid_ci$erakond[1:9]) +
  labs(x = "Erakond",
       y = "Osakaal") +
  theme_light() 
```

---

### Paneme reitingud koos usalduspiiridega joonisele

---

### Usaldusvahemike võrdlemine

- Usaldusvahemikud kattuvad omajagu

- Pelgalt usaldusevahemike kattumise vaatlemisel võiks öelda, et RE reiting võib tõepoolest olla tõusnud, aga võib ka olla, et valimite juhuslikkuse tõttu on reitingud olnud kahel kuul samad või üsnagi sarnased

- Sama saaks öelda ka KE kohta

- Siiski, sarnaselt keskmiste võrldemisega tuleks siingi võtta arvesse osakaalude erinevuse usaldusvahemikku (st RE reitingumuutuse usaldusvahemikku ja KE reitingumuutuse usaldusvahemikku)

- Kuidas osakaalude erinevuse usalduspiire arvutada?

---

### Osakaalude erinevuse usalduspiirid

- Kuidas osakaalude vahe usalduspiire arvutada?

- Erinev arvutusvalem sõltuvalt sellest, kas osakaalud on

- kahest erinevast jaotusest (kahest teineteisest sõltumatust kogumist) 
  
  - või samast jaotusest (osakaalud on osa ühest tervikust)

- Nt erakondade reitingute uurimise puhul:

- kas uurime erinevust ühe erakonna reitingutes kahel erineval ajahetkel 
  
  - või erakondade reitingute erinevusi samal ajahetkel

- Esimene juhtum: kahe osakaalu *z*-test (sõltumatute kogumite *z*-test)

---

### Kahe osakaalu *z*-test

`$$Z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$$`

- `$\hat{p}_1$` -- esimese grupi osakaal esimesest valimist `$(n_1)$`

- `$\hat{p}_2$` -- teise grupi osakaal teisest valimist `$(n_2)$`

- `$\hat{p}$` -- mõlema grupi osakaal mõlema valimi peale kokku: `$\frac{\hat{p}_1 + \hat{p}_2}{n_1+n_2}$`

- `$n_1$` -- esimese valimi suurus

- `$n_2$` -- teise valimi suurus

---

### Osakaalude erinevus kahe osakaalu *z*-testi põhjal

`$$Z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$$`

- Kasutame funktsiooni `prop.test`, millele on vaja eraldi argumentidena ette anda vektor indiviidide arvudega kummaski grupis `$c(\hat{p}_1 n_1, \hat{p}_2 n_2)$` ja valimimahtude vektor `$c(n_1, n_2)$`.

- Osakaalude vektor:

```r
ind_arv_gruppides <- reiting %>% 
  filter(erakond == "RE") %>% 
  pull(poolehoid_n)

ind_arv_gruppides
```

```
## [1] 231 273
```

---

### Osakaalude erinevus kahe osakaalu *z*-testi põhjal

`$$Z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$$`

Arvutame, kui palju vastajaid oli augustis ja septembris

```r
n1 <- reiting %>% 
  filter(kuu == "August") %>% 
  pull(poolehoid_n) %>% 
  sum()
n1
```

```
## [1] 1003
```

```r
n2 <- reiting %>% 
  filter(kuu == "September") %>% 
  pull(poolehoid_n) %>% 
  sum()
n2
```

```
## [1] 1009
```

---

### Osakaalude erinevus kahe osakaalu *z*-testi põhjal

`$$Z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$$`

```r
prop.test(ind_arv_gruppides, c(n1, n2), correct = FALSE)
```

```
## 
## 	2-sample test for equality of proportions without continuity correction
## 
## data:  ind_arv_gruppides out of c(n1, n2)
## X-squared = 4.3416, df = 1, p-value = 0.03719
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  -0.078075315 -0.002436371
## sample estimates:
##    prop 1    prop 2 
## 0.2303091 0.2705649
```

---

### Osakaalude erinevus kahe osakaalu *z*-testi põhjal

- Tulemus näitab, et usaldusnivool 95% saab RE reitingumuutust väita küll

- usaldusvahemik ei sisalda nulli, kuigi, tõsi küll, ülemine ots ei ole nullist kuigi kaugel
  
  - sama näitab teststatistiku olulisuse tõenäosus, mis on väiksem 0,05-st (usaldusnivoole 95% vastavast olulisuse nivoost)

- `prop.test` annab meile küll hii-ruut-statistiku, mitte *Z*, aga olulisuse tõenäosus on sama

---

### Harjutus 6

- Kas samal ajal kui RE reiting augusti ja septembri võrdluses tõusis, saame väita, et KE reiting langes?

---

### Sama jaotuse osakaalude erinevuse usalduspiirid

- Kuidas arvutada osakaalude erinevuse usalduspiire, kui osakaalud on pärit samast protsentjaotusest?

- Nt reitingute erinevus samal ajahetkel

- Osakaalude erinevuse usalduspiiride valem (Scott ja Seber 1983):

`$$p_1 - p_2 >= \hat{p}_1 - \hat{p}_2 - z_{1 - \frac{\alpha}{2}}\sqrt{\frac{\hat{p}_1+\hat{p}_2-(\hat{p}_1-\hat{p}_2)^2}{n}}$$`

`$$p_1 - p_2 <= \hat{p}_1 - \hat{p}_2 + z_{1 - \frac{\alpha}{2}}\sqrt{\frac{\hat{p}_1+\hat{p}_2-(\hat{p}_1-\hat{p}_2)^2}{n}}$$`

- `$p_1, p_2$` -- esimese ja teise grupi osakaal populatsioonis

- `$\hat{p}_1, \hat{p}_2$` -- esimese ja teise grupi osakaal valimis

- `$z_{1 - \frac{\alpha}{2}}$` -- standardiseeritud normaaljaotuse `$1 - \frac{\alpha}{2}$`-kvantiil

- `$n$` -- valimi suurus

---

### Sama jaotuse osakaalude erinevuse usalduspiirid

- Uudise pealkiri oli, et RE tõusis septembris taas populaarseimaks parteiks

- Kas saame siiski uuringu andmetel öelda, et RE ja KE septembrireitingud erinevad ka populatsioonis (eeldades, et täisarvulised protsendid on täpsed ja et uuring põhineb lihtsal juhuvalimil)?

- Täpsemalt, kas usaldusnivool 95% on alust väita RE ja KE septembrireitingute erinevust?

- Arvutame reitingute vahe usalduspiirid

- Selle jaoks on vaja RE ja KE septembrikuu reitinguid ja kõigi vastajate arvu septembris (viimane on eelnevalt arvutatud objektis `n2`).

---

### RE ja KE septembrikuu reitingud

- RE septembrikuu reiting

```r
p1 <- reiting %>% 
  filter(erakond == "RE" & kuu == "September") %>% 
  pull(poolehoid)
```

- KE septembrikuu reiting

```r
p2 <- reiting %>% 
  filter(erakond == "KE" & kuu == "September") %>% 
  pull(poolehoid)
```

---

### RE ja KE septembrireitingute erinevuse usalduspiirid

- Reitingute erinevus valimis

```r
p1 - p2
```

```
## [1] 0.03
```

- Alumine usalduspiir: `$p_1 - p_2 >= \hat{p}_1 - \hat{p}_2 - z_{1 - \frac{\alpha}{2}}\sqrt{\frac{\hat{p}_1+\hat{p}_2-(\hat{p}_1-\hat{p}_2)^2}{n}}$`

```r
p1 - p2 - 1.96 * sqrt((p1 + p2 - (p1 - p2)^2) / n2) 
```

```
## [1] -0.01402628
```

- Ülemine usalduspiir: `$p_1 - p_2 <= \hat{p}_1 - \hat{p}_2 + z_{1 - \frac{\alpha}{2}}\sqrt{\frac{\hat{p}_1+\hat{p}_2-(\hat{p}_1-\hat{p}_2)^2}{n}}$`

```r
p1 - p2 + 1.96 * sqrt((p1 + p2 - (p1 - p2)^2) / n2) 
```

```
## [1] 0.07402628
```

- Mida nendest usalduspiiridest järeldada saame? (NB! Andmed on 2020. aasta septembri kohta)

---

### Harjutus 7

- KE ja RE reitingute erinevust 2020 septembris väita ei saanud. Kas 2020 augustis oli RE edumaa siiski statistilisest veast suurem, st sai väita RE kõrgemat reitingut KE-st?